How to split mesh into several smaller ones with respecting UV map?
I believe I have a way. This is what I did, and it worked perfectly.


Shift + A > Mesh > Plane.
Go to the Properties panel > Object Modifiers tab and add a Subdivision Surface modifier. Use these settings:


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Key Tab to enter Edit Mode, U to open the Unwrap menu, and select Unwrap.
Split the 3D Viewport and make the new panel a UV Image Editor. Open your image in the editor.





Tab out of Edit Mode.
Add a Displace modifier. Click the new texture button. Use these settings:


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Go to the Properties panel > Textures tab. Load in your image from the dropdown menu.


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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;You should now see your texture displayed on the plane in the 3D Viewport. Note that you
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;may have to adjust the Displace modifier > Midlevel: value for your texture to sit right near the
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;origin point. I have to use 0.836.

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Tab into Edit Mode. Key W to open the Specials menu and select Subdivide. Key T to toggle open/close the Toolshelf. Change the Number of Cuts to 3.


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In the UV Image Editor, key S, then .25. Additional UV Editing may be required, but this should start you.
Key Ctrl + Tab to open the Mesh Select Mode menu and either key 2 or select Edge.
Select these edges:


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Key V to Rip the edges.
Select these edges:


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Key V to Rip the edges.
Key Ctrl + Tab to open the Mesh Select Mode menu and either key 3 or select Face.
Key A once to select everything.
Key P to open the Separate menu and select By Loose Parts.
Tab out of Edit Mode.


Note that one reason the separated planes may not look the same as one intact planes is that the separation provided each plane with an additional two subsurf levels, effectively. You would have to turn down the subsurf to 4 on each midget plane to get the midget planes to look the same as one large plane.

Here is the .blend file: